Question

question 24 of 44
what is the measure of arc ( overarc{wx} ) in the diagram below?
diagram: circle with points t, r, x, w; lines st (tangent/secant) and sr (secant); angle at s is 45°, arc tr is 115°; options: a. 116° b. 26° c. 71° d. 45°; \click here for long description\

Explanation:

Step1: Recall the tangent - secant angle theorem

The measure of an angle formed by a tangent and a secant (or two secants, or two tangents) outside a circle is equal to half the difference of the measures of the intercepted arcs. The formula is: if an angle outside the circle is formed by a tangent and a secant, then \(m\angle S=\frac{1}{2}(m\widehat{TR}-m\widehat{WX})\), where \(m\angle S\) is the measure of the angle outside the circle, \(m\widehat{TR}\) is the measure of the larger intercepted arc, and \(m\widehat{WX}\) is the measure of the smaller intercepted arc.

We know that \(m\angle S = 45^{\circ}\) and \(m\widehat{TR}=116^{\circ}\). Let \(m\widehat{WX}=x\).

Step2: Substitute the known values into the formula

Substitute into the formula \(45^{\circ}=\frac{1}{2}(116^{\circ}-x)\).

First, multiply both sides of the equation by 2 to get rid of the fraction:
\(2\times45^{\circ}=116^{\circ}-x\)
\(90^{\circ}=116^{\circ}-x\)

Step3: Solve for \(x\)

Rearrange the equation to solve for \(x\):
\(x = 116^{\circ}- 90^{\circ}\)
\(x=26^{\circ}\)

Answer:

B. 26°